Fractional Breakup Velocities

• Fractional breakup velocities are threshold speeds defined by nonlocal and fractional dynamics, governing fragmentation in systems like quantum scattering, asteroid collisions, and droplet breakup.
• Theoretical models use fractional Schrödinger operators, power-law scaling, and dimensionless velocity ratios to ensure minimal propagation speeds and predictable fragment ejection patterns.
• These insights unify diverse phenomena—from asymptotic quantum behavior to viscous fluid dynamics and nonlinear soliton collisions—informing both experimental designs and numerical simulations.

Fractional breakup velocities refer to characteristic velocity thresholds or distributions controlling the fragmentation of an aggregate or wavepacket in systems exhibiting non-integer order (fractional) dynamics or scaling. These velocities appear prominently in nonlocal dispersive quantum systems, astrophysical collision ejecta, viscous and inertial droplet fragmentation, as well as in nonlinear soliton dynamics where limiting velocities, scaling laws, or probabilistic thresholds dictate breakup transitions. In all such contexts, the fractionality may appear as explicit nonlocal operators, explicit power-law scaling, or dimensionless ratios encoding the effect of system heterogeneity or damping on breakup phenomena.

1. Fractional Minimal Velocity Bounds in Quantum Scattering

The framework for minimal velocity bounds in quantum scattering with fractional dynamical generators is formalized through fractional Schrödinger-type operators

$$H_p = V_p(|D|^2) + V(x), \quad \text{with} \quad V_p(|D|^2)\varphi(x) = (2\pi)^{-n}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{-i(x-y)\cdot\xi}\left[(|\xi|^2+1)^{p}-1\right]\varphi(y)dyd\xi,$$

where $0 < p < 1$ is the fractional power and $V(x)$ includes Coulomb-type singularities, short-range, and long-range decay components. The group velocity is given by $\nabla_\xi V_p(|\xi|^2)=2p(|\xi|^2+1)^{p-1}\xi$ (Ishida, 2020).

The minimal velocity bound (Theorem 1.4), stated rigorously as

$$\int_1^\infty \|\chi_{|x|<\theta t}f(H_p)e^{-itH_p}\psi\|_{L^2}^2 dt \leq C\|\psi\|_{L^2}^2,$$

implies that the time-evolved wavepacket cannot remain inside a shrinking spatial cone for long times, ensuring an asymptotic outward motion at a strictly positive minimal speed, even for non-integer $p$. This property plays a central role in establishing asymptotic completeness and excludes arbitrarily slow or non-dispersive breakup channels for energy-localized quantum states, with the fractional group-velocity modifying the constants but preserving positivity.

These propagation estimates are extended to many-body and long-range fractional systems, often requiring modified wave operators in the analysis. The methods rely crucially on spectral projectors, weighted commutator estimates, the Helffer–Sjöstrand functional calculus, and domain techniques specific to the fractional setting.

2. Scaling Laws for Breakup Velocities in Asteroid Collisions

In planetary science, the concept of fractional breakup velocities is exemplified by the reconstruction of the ejection-speed field of asteroid families—most notably, the Koronis family. Here, the velocity field of fragments ejected during a high-energy collision follows a power-law scaling with respect to fragment diameter $D$:

$$v(D) = K D^{-\alpha},\qquad\alpha = 1.0\pm 0.15,\quad K = 80\pm12~\mathrm{m/s},\, (2 < D < 10~\mathrm{km}),$$

where $v(D)$ is the root-mean-square ejection velocity inferred from the observed inclination spread after correction for dynamical diffusion and Yarkovsky/YORP perturbations (Carruba et al., 2016).

The perpendicular component ($v_W$) of the ejection velocity is particularly amenable to deconvolution using the Gauss form of the variation of orbital elements, and the inferred breakup velocities can occupy a wide range compared to the escape velocity $V_{\mathrm{esc}}$ of the parent body:

$0.4 \leq \frac{v_W}{V_{\mathrm{esc}}} \leq 1.5,$

demonstrating that breakup velocities are typically a substantial fraction of $V_{\mathrm{esc}}$. The scaling $v(D)\propto D^{-1}$ encapsulates the “fractional” aspect, where smaller fragments are ejected at systematically higher velocities.

3. Relative Fractional Breakup Velocities in Viscous Droplet Fragmentation

The transition between morphological breakup regimes in viscous droplets in airflow is governed by dimensionless velocity ratios. The critical velocity at which a droplet undergoes breakup, relative to its initial value $U_0$, is modified from the inviscid expectation by viscous drag, which accumulates during the initial flattening:

$$U_{\rm crit}/U_0 = \frac{1}{1+1.8\,\mathrm{Oh}~\mathrm{We}_c^{-1/2}},$$

where Oh is the Ohnesorge number, We$_c$ is the critical Weber number for a given regime, and the numerator follows from drag-loss models and high-speed imaging (Xu et al., 19 Apr 2025).

The curves of $U_{\rm crit}/U_0$ versus Ohnesorge number for each regime boundary show that, for low Oh, the fractional critical velocity approaches unity (inviscid case), while for Oh~2, the required airflow speed at breakup can be 1.3–1.6 times the inviscid threshold, contingent on the particular transition. This “fractional” velocity captures the cumulative dissipative effects on breakup onset and forms a continuum that connects classical regime maps with hydrodynamic instability theory incorporating viscosity.

4. Breakup Thresholds and Probabilities in Obstacle-Induced Droplet Fragmentation

In confined microfluidic or 2D obstacle-flow geometries, the breakup transition for droplets impacting a circular obstacle is universally characterized by a nondimensional "breakup number" (Bk), which—at fixed geometry—scales as

$$\mathrm{Bk} \approx 28\,\mathrm{Ca} \left(\frac{A}{R^2}\right)\left(\frac{z}{R}\right) S^{4/3},$$

where Ca is the Capillary number ($\mu v/\gamma$), $A$ is droplet area, $R$ is obstacle radius, $z$ is chamber height, and $S$ characterizes impact symmetry ($0\le S\le1$) (Meer et al., 23 Dec 2025).

The probability of breakup, for both experiments and simulations, follows a near-universal sigmoid as a function of Bk:

$$P_{\rm breakup}(\mathrm{Bk}) = \frac{\mathrm{Bk}}{1+\mathrm{Bk}},$$

with the transition from almost no breakup ($\mathrm{Bk}\ll1$) to certain breakup ($\mathrm{Bk}\gg1$) compressed into a window near Bk$\approx1$, corresponding to capillary numbers and velocities that are “fractional” relative to the geometric and physicochemical system parameters. The $S^{4/3}$ scaling reveals heightened sensitivity of the breakup threshold to collision symmetry.

5. Velocity Distributions from Rim Retraction in Bag Breakup

During bag breakup events, subgrid-scale models predict the velocity distribution of emergent droplets via the Taylor–Culick rim retraction mechanism. The velocity of a drop ejected from a retracting film of local thickness $h$ is given by

$U_{TC} = \sqrt{2\sigma/(\rho_\ell h)},$

where $\sigma$ is surface tension and $\rho_\ell$ is liquid density. The fractional breakup velocity $F(d)$ of a drop of diameter $d$ is then

$$F(d) \equiv \frac{|\mathbf{u}_{\rm drop}|}{U_0} \approx \frac{d_0}{d}\mathrm{We}^{-1/2},$$

given $h\propto d^2/d_0$ and $\mathrm{We}=\rho_g U_0^2 d_0/\sigma$, where $d_0$ is the parent drop diameter and $U_0$ is the bulk velocity (Han et al., 2024).

This framework yields a negative correlation between daughter drop size and normalized velocity, consistent with experimental data: smaller fragments attain higher velocities, and the entire distribution is “fractionally” determined relative to the initial velocity. The theoretical velocity–size PDF thus emerges from sampling the empirically calibrated gamma law for size together with the Taylor–Culick scaling for breakup speed.

6. Maximum and Critical Velocities for Soliton Breakup in Fractional Nonlinear Media

For two-dimensional solitons governed by the fractional nonlinear Schrödinger equation with cubic–quintic nonlinearity,

$$i\frac{\partial u}{\partial z} = \frac{1}{2}(-\partial_x^2-\partial_y^2)^{\alpha/2}u - |u|^2u + |u|^4u,$$

the breaking of Galilean invariance for Lévy index $\alpha < 2$ introduces a well-defined finite maximum velocity $c_{\rm max}(\alpha,k)$ for moving (“tilted”) solitons (Mayteevarunyoo et al., 2024). Numerically determined values include

$$c_{\rm max}(1.5, 0.135) \approx 0.60, \quad c_{\rm max}(1.0, 0.09) \approx 0.49,$$

for fundamental solitons of fixed power. Collisions between solitons exhibit two critical velocity thresholds $v_{c1}$ (below which merger occurs) and $v_{c2}$ (above which quartet breakup or destruction is triggered), with all three thresholds determined numerically and lying well below $c_{\rm max}(\alpha, k)$.

The non-integer order of the dispersive operator reduces these maximal velocities relative to the $\alpha=2$ case (where $c_{\rm max}\to\infty$ by Galilean boost), thereby enforcing a “fractional” velocity limit for persistent soliton transport or collision-induced breakup.

7. Synthesis and Physical Significance

Fractional breakup velocities emerge as velocity thresholds, scaling laws, or distributions shaped by non-integer dynamical exponents, nonlocal operators, or stochastic-geometry factors. They appear in quantum systems where they bound the minimal outward speed of scattering channels and ensure completeness, in astrophysical fragmentation with size-dependent ejection speeds relative to escape velocity, in fluid dynamics as critical or probabilistic thresholds for droplet breakup, and in nonlinear optics as limiting velocities for soliton propagation and fragmentation.

Across these domains, the “fractionality” reflects the nontrivial dependence of breakup or dispersal on system-intrinsic exponents, often arising from long-range interactions, nonlocal dispersion, or geometry-dependent balance of inertia, dissipation, and restoring forces. These results provide foundations for predicting and quantifying the onset, rate, and probabilistic likelihood of breakup phenomena in complex media and under nonlocal dynamics, directly informing theoretical, experimental, and computational studies in multiple branches of physics, astrophysics, and engineering (Ishida, 2020, Carruba et al., 2016, Xu et al., 19 Apr 2025, Meer et al., 23 Dec 2025, Han et al., 2024, Mayteevarunyoo et al., 2024).